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Questions in mathematics

📝 Answered - Prove that if $d \leq n$, then $S_n$ contains elements of order $d$. For every positive integer $n$, find an element of order $n$ in $S_{ N }$.

📝 Answered - Simplify the complex fraction. [tex]$\frac{\frac{4}{p+4}-1}{\frac{4}{p+4}+1}$[/tex]

📝 Answered - Find the limit. Use a graphing utility to verify your result. (Hint: Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) [tex]\lim _{x \rightarrow-\infty}\left(4 x+\sqrt{16 x^2-x}\right)[/tex]

📝 Answered - Which will result in a perfect square trinomial? $(3 x-5)(3 x-5)$ $(3 x-5)(5-3 x)$ $(3 x-5)(3 x+5)$ $(3 x-5)(-3 x-5)$

📝 Answered - To test the effect of a physical fitness course on one's physical ability, the number of sit-ups that a person could do in one minute, both before and after the course, was recorded. Ten individuals are randomly selected to participate in the course. The results are displayed in the following table. Using this data, find the $99 \%$ confidence interval for the true difference in the number of sit-ups each person can do before and after the course. Assume that the numbers of sit-ups are normally distributed for the population both before and after completing the course. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline Sit-ups before & 46 & 53 & 36 & 48 & 40 & 33 & 47 & 37 & 47 & 24 \\ \hline Sit-ups after & 55 & 59 & 40 & 57 & 56 & 49 & 60 & 39 & 56 & 26 \\ \hline \end{tabular} Step 3 of 4: Calculate the margin of error to be used in constructing the confidence interval. Round your answer to six decimal places.

📝 Answered - [tex]\frac{12}{x^7} \times \frac{7}{x^{12}}[/tex]

📝 Answered - At 6 a.m., the temperature was $-8^{\circ} C$. By noon, the temperature had risen $6^{\circ} C$. The temperature at 6 p.m. was $-10^{\circ} C$. Which integer represents the temperature change between noon and 6 p.m.? A. $-24$ B. $-12$ C. $-8$ D. $-4

📝 Answered - Marlena solved the equation $2 x+5=-10-x$. Her steps are shown below. $2 x+5=-10-x$ 1. $3 x+5=-10$ 2. $3 x=-15$ 3. $x=-5$ Use the drop-down menus to justify Marlena's work in each step of the process. Step 1: $\square$ Step 2: $\square$ Step 3: $\square$

📝 Answered - Simplify the complex fraction. $\frac{\frac{4}{x+4}+2}{\frac{10}{x+4}-2}$

📝 Answered - Use a graphing utility to complete the table and estimate the limit as [tex]$x$[/tex] approaches infinity. (Round your answers to five decimal places.) [tex]$f(x)=x+\left(\frac{1}{6 x}\right)$[/tex] | x | [tex]$10^0$[/tex] | [tex]$10^1$[/tex] | [tex]$10^2$[/tex] | [tex]$10^3$[/tex] | [tex]$10^4$[/tex] | [tex]$10^5$[/tex] | [tex]$10^6$[/tex] | |---|---|---|---|---|---|---|---| | f(x) | | | | | | | Use a graphing utility to graph the function and estimate the limit. Find the limit analytically and compare your results with the estimates. What is the exact limit? (If an answer does not exist, enter DNE.) [tex]$\lim _{x \rightarrow \infty} f(x)=[/tex]